Abstract

We developed an approach to the blind multichannel reconstruction of high-resolution images. This approach is based on breaking the image reconstruction problem into three consecutive steps: a blind multichannel restoration, a wavelet-based image fusion, and a maximum entropy image interpolation. The blind restoration step depends on estimating the two-dimensional (2-D) greatest common divisor (GCD) between each observation and a combinational image generated by a weighted averaging process of the available observations. The purpose of generating this combinational image is to get a new image with a higher signal-to-noise ratio and a blurring operator that is a coprime with all the blurring operators of the available observations. The 2-D GCD is then estimated between the new image and each observation, and thus the effect of noise on the estimation process is reduced. The multiple outputs of the restoration step are then applied to the image fusion step, which is based on wavelets. The objective of this step is to integrate the data obtained from each observation into a single image, which is then interpolated to give an enhanced resolution image. A maximum entropy algorithm is derived and used in interpolating the resulting image from the fusion step. Results show that the suggested blind image reconstruction approach succeeds in estimating a high-resolution image from noisy blurred observations in the case of relatively coprime unknown blurring operators. The required computation time of the suggested approach is moderate.

© 2005 Optical Society of America

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    [CrossRef]
  18. M. K. Ng, N. K. Bose, “Mathematical analysis of super-resolution methodology,” IEEE Signal Process. Mag. 20, 62–74 (2003).
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  19. M. Vega, J. Mateos, R. Molina, A. K. Katsagegelos, “Bayesian parameter estimation in image reconstruction from sub-sampled blurred observations,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, 2003), pp. 1655–1657.
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  25. S. U. Pillai, B. Liang, “Blind image deconvolution using a robust GCD approach,” IEEE Trans. Image Process. 8, 295–301 (1999).
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  29. H. Li, B. S. Manjunath, S. K. Mitra, “Multi-sensor image fusion using the wavelet transform,” in Proceedings of IEEE International Conference on Image Processing (ICIP’94), (Institute of Electrical and Electronics Engineers, 1994), pp. 51–53.
  30. G. Piella, H. Heijmans, “Multiresolution image fusion guided by a multimodal segmentation,” in Proceedings of ACIVS 2002 (Advanced Concepts for Intelligent Vision Systems), Ghent, Belgium, September2002, pp. 175–182.
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    [CrossRef]
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    [CrossRef]
  33. S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Efficient implementation of image interpolation as an inverse problem,” Digital Signal Process. 15, 137–152 (2005).
    [CrossRef]
  34. S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Sectioned implementation of regularized image interpolation,” in 46th Proceedings of the IEEE Midwest Symposium on Circuits and Systems (Institute of Electrical and Electronics Engineers, 2003), pp. 656–659.
    [CrossRef]
  35. J. H. Shin, J. H. Jung, J. K. Paik, “Regularized iterative image interpolation and its application to spatially scalable coding,” IEEE Trans. Consumer Electron. 44, 1042–1047 (1998).
    [CrossRef]

2005

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Efficient implementation of image interpolation as an inverse problem,” Digital Signal Process. 15, 137–152 (2005).
[CrossRef]

T. J. Schulz, D. G. Voelz, “Signal recovery from autocorrelation and cross-correlation data,” J. Opt. Soc. Am. A 22, 616–624 (2005).
[CrossRef]

2003

D. Capel, A. Zisserman, “Computer vision applied to super resolution,” IEEE Signal Process. Mag. 20(3), 75–86 (2003).
[CrossRef]

R. Nakagaki, A. K. Katsaggelos, “A VQ-based blind image restoration algorithm,” IEEE Trans. Image Process. 12, 1044–1053 (2003).
[CrossRef]

F. Sroubek, J. Flusser, “Multichannel blind iterative image restoration,” IEEE Trans. Image Process. 12, 1094–1106 (2003).
[CrossRef]

S. C. Park, M. K. Park, M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21–36 (2003).
[CrossRef]

C. A. Segall, R. Molina, A. K. Katsaggelos, “High-resolution images from low-resolution compressed video,” IEEE Signal Process. Mag. 20(3), 37–48 (2003).
[CrossRef]

D. Rajan, S. Chandhuri, M. V. Joshi, “Multi-objective super resolution: concepts and examples,” IEEE Signal Process. Mag. 20(3), 49–61 (2003).
[CrossRef]

M. K. Ng, N. K. Bose, “Mathematical analysis of super-resolution methodology,” IEEE Signal Process. Mag. 20, 62–74 (2003).
[CrossRef]

2001

N. Nguyen, P. Milanfar, G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Process. 10, 573–583 (2001).
[CrossRef]

W. Y. V. Leung, P. J. Bones, “Statistical interpolation of sampled images,” Opt. Eng. 40, 547–553 (2001).
[CrossRef]

1999

M. Elad, A. Feuer, “Superresolution restoration of an image sequence: adaptive filtering approach,” IEEE Trans. Image Process. 8, 387–395 (1999).
[CrossRef]

S. U. Pillai, B. Liang, “Blind image deconvolution using a robust GCD approach,” IEEE Trans. Image Process. 8, 295–301 (1999).
[CrossRef]

M. Fiddy, R. Millane, “Signal recovery and synthesis,” J. Opt. Soc. Am. A 16, 1742 (1999).
[CrossRef]

G. Harikumar, Y. Bresler, “Perfect blind restoration of images blurred by multiple filters: theory and efficient algorithms,” IEEE Trans. Image Process. 8, 202–219 (1999).
[CrossRef]

1998

J. H. Shin, J. H. Jung, J. K. Paik, “Regularized iterative image interpolation and its application to spatially scalable coding,” IEEE Trans. Consumer Electron. 44, 1042–1047 (1998).
[CrossRef]

1997

H. T. Pai, A. C. Bovik, “Exact multichannel blind image restoration,” IEEE Signal Process. Lett. 4, 217–220 (1997).
[CrossRef]

M. Elad, A. Feuer, “Restoration of a single superresolution image from several blurred noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646–1658 (1997).
[CrossRef]

P. E. Eren, M. I. Sezan, M. Tekalp, “Robust, object-based high-resolution image reconstruction from low-resolution video,” IEEE Trans. Image Process. 6, 1446–1451 (1997).
[CrossRef] [PubMed]

1993

S. P. Kim, W. Y. Su, “Recursive high resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534–539 (1993).
[CrossRef]

1990

S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive Reconstruction of high resolution image from noisy under-sampled multiframes,” IEEE Trans. Acoust. Speech, Signal Process. 38, 1013–1027 (1990).
[CrossRef]

1987

Abd El-Samie, F. E.

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Efficient implementation of image interpolation as an inverse problem,” Digital Signal Process. 15, 137–152 (2005).
[CrossRef]

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Sectioned implementation of regularized image interpolation,” in 46th Proceedings of the IEEE Midwest Symposium on Circuits and Systems (Institute of Electrical and Electronics Engineers, 2003), pp. 656–659.
[CrossRef]

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Optimization of image interpolation as an inverse problem using the LMMSE algorithm,” in Proceedings of 12th IEEE Mediterranean Electrotechnical Conference (Institute of Electrical and Electronic Engineers, 2004), pp. 247–250.
[CrossRef]

Andrews, H. C.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

Baker, S.

S. Baker, T. Kanade, “Super-resolution: reconstruction or recognition,” in Proceedings of the IEEE-EURASIP Workshop On Nonlinear Signal and Image Processing (NSIP 2001) (Institute of Electrical and Electronics Engineers, 2001), paper SS1.3.

Barrett, H. H.

H. H. Barrett, K. J. Myers, Foundations of Image Science (Wiley, 2004), Chap. 15.

Bones, P. J.

W. Y. V. Leung, P. J. Bones, “Statistical interpolation of sampled images,” Opt. Eng. 40, 547–553 (2001).
[CrossRef]

Bose, N. K.

M. K. Ng, N. K. Bose, “Mathematical analysis of super-resolution methodology,” IEEE Signal Process. Mag. 20, 62–74 (2003).
[CrossRef]

S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive Reconstruction of high resolution image from noisy under-sampled multiframes,” IEEE Trans. Acoust. Speech, Signal Process. 38, 1013–1027 (1990).
[CrossRef]

Bovik, A. C.

H. T. Pai, A. C. Bovik, “Exact multichannel blind image restoration,” IEEE Signal Process. Lett. 4, 217–220 (1997).
[CrossRef]

Bresler, Y.

G. Harikumar, Y. Bresler, “Perfect blind restoration of images blurred by multiple filters: theory and efficient algorithms,” IEEE Trans. Image Process. 8, 202–219 (1999).
[CrossRef]

Capel, D.

D. Capel, A. Zisserman, “Computer vision applied to super resolution,” IEEE Signal Process. Mag. 20(3), 75–86 (2003).
[CrossRef]

Chandhuri, S.

D. Rajan, S. Chandhuri, M. V. Joshi, “Multi-objective super resolution: concepts and examples,” IEEE Signal Process. Mag. 20(3), 49–61 (2003).
[CrossRef]

Dessouky, M. I.

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Efficient implementation of image interpolation as an inverse problem,” Digital Signal Process. 15, 137–152 (2005).
[CrossRef]

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Sectioned implementation of regularized image interpolation,” in 46th Proceedings of the IEEE Midwest Symposium on Circuits and Systems (Institute of Electrical and Electronics Engineers, 2003), pp. 656–659.
[CrossRef]

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Optimization of image interpolation as an inverse problem using the LMMSE algorithm,” in Proceedings of 12th IEEE Mediterranean Electrotechnical Conference (Institute of Electrical and Electronic Engineers, 2004), pp. 247–250.
[CrossRef]

Elad, M.

M. Elad, A. Feuer, “Superresolution restoration of an image sequence: adaptive filtering approach,” IEEE Trans. Image Process. 8, 387–395 (1999).
[CrossRef]

M. Elad, A. Feuer, “Restoration of a single superresolution image from several blurred noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646–1658 (1997).
[CrossRef]

El-Khamy, S. E.

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Efficient implementation of image interpolation as an inverse problem,” Digital Signal Process. 15, 137–152 (2005).
[CrossRef]

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Optimization of image interpolation as an inverse problem using the LMMSE algorithm,” in Proceedings of 12th IEEE Mediterranean Electrotechnical Conference (Institute of Electrical and Electronic Engineers, 2004), pp. 247–250.
[CrossRef]

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Sectioned implementation of regularized image interpolation,” in 46th Proceedings of the IEEE Midwest Symposium on Circuits and Systems (Institute of Electrical and Electronics Engineers, 2003), pp. 656–659.
[CrossRef]

Eren, P. E.

P. E. Eren, M. I. Sezan, M. Tekalp, “Robust, object-based high-resolution image reconstruction from low-resolution video,” IEEE Trans. Image Process. 6, 1446–1451 (1997).
[CrossRef] [PubMed]

Feuer, A.

M. Elad, A. Feuer, “Superresolution restoration of an image sequence: adaptive filtering approach,” IEEE Trans. Image Process. 8, 387–395 (1999).
[CrossRef]

M. Elad, A. Feuer, “Restoration of a single superresolution image from several blurred noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646–1658 (1997).
[CrossRef]

Fiddy, M.

Fienup, J. R.

Flusser, J.

F. Sroubek, J. Flusser, “Multichannel blind iterative image restoration,” IEEE Trans. Image Process. 12, 1094–1106 (2003).
[CrossRef]

Frieden, B. R.

B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Filtering, T. S. Huang, ed. (Springer, 1975), Vol. 6, pp. 177–248.
[CrossRef]

Golub, G.

N. Nguyen, P. Milanfar, G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Process. 10, 573–583 (2001).
[CrossRef]

Hadhoud, M. M.

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Efficient implementation of image interpolation as an inverse problem,” Digital Signal Process. 15, 137–152 (2005).
[CrossRef]

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Optimization of image interpolation as an inverse problem using the LMMSE algorithm,” in Proceedings of 12th IEEE Mediterranean Electrotechnical Conference (Institute of Electrical and Electronic Engineers, 2004), pp. 247–250.
[CrossRef]

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Sectioned implementation of regularized image interpolation,” in 46th Proceedings of the IEEE Midwest Symposium on Circuits and Systems (Institute of Electrical and Electronics Engineers, 2003), pp. 656–659.
[CrossRef]

Harikumar, G.

G. Harikumar, Y. Bresler, “Perfect blind restoration of images blurred by multiple filters: theory and efficient algorithms,” IEEE Trans. Image Process. 8, 202–219 (1999).
[CrossRef]

Heijmans, H.

G. Piella, H. Heijmans, “Multiresolution image fusion guided by a multimodal segmentation,” in Proceedings of ACIVS 2002 (Advanced Concepts for Intelligent Vision Systems), Ghent, Belgium, September2002, pp. 175–182.

Hunt, B. R.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

Joshi, M. V.

D. Rajan, S. Chandhuri, M. V. Joshi, “Multi-objective super resolution: concepts and examples,” IEEE Signal Process. Mag. 20(3), 49–61 (2003).
[CrossRef]

Jung, J. H.

J. H. Shin, J. H. Jung, J. K. Paik, “Regularized iterative image interpolation and its application to spatially scalable coding,” IEEE Trans. Consumer Electron. 44, 1042–1047 (1998).
[CrossRef]

Kanade, T.

S. Baker, T. Kanade, “Super-resolution: reconstruction or recognition,” in Proceedings of the IEEE-EURASIP Workshop On Nonlinear Signal and Image Processing (NSIP 2001) (Institute of Electrical and Electronics Engineers, 2001), paper SS1.3.

Kang, M. G.

S. C. Park, M. K. Park, M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21–36 (2003).
[CrossRef]

Katsagegelos, A. K.

M. Vega, J. Mateos, R. Molina, A. K. Katsagegelos, “Bayesian parameter estimation in image reconstruction from sub-sampled blurred observations,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, 2003), pp. 1655–1657.

Katsaggelos, A. K.

C. A. Segall, R. Molina, A. K. Katsaggelos, “High-resolution images from low-resolution compressed video,” IEEE Signal Process. Mag. 20(3), 37–48 (2003).
[CrossRef]

R. Nakagaki, A. K. Katsaggelos, “A VQ-based blind image restoration algorithm,” IEEE Trans. Image Process. 12, 1044–1053 (2003).
[CrossRef]

Kim, S. P.

S. P. Kim, W. Y. Su, “Recursive high resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534–539 (1993).
[CrossRef]

S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive Reconstruction of high resolution image from noisy under-sampled multiframes,” IEEE Trans. Acoust. Speech, Signal Process. 38, 1013–1027 (1990).
[CrossRef]

Leung, W. Y. V.

W. Y. V. Leung, P. J. Bones, “Statistical interpolation of sampled images,” Opt. Eng. 40, 547–553 (2001).
[CrossRef]

Levi, A.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery—Theory and Applications, H. Stark, ed.(Academic, 1987), Chap. 8, pp. 277–320.

Li, H.

H. Li, B. S. Manjunath, S. K. Mitra, “Multi-sensor image fusion using the wavelet transform,” in Proceedings of IEEE International Conference on Image Processing (ICIP’94), (Institute of Electrical and Electronics Engineers, 1994), pp. 51–53.

Liang, B.

S. U. Pillai, B. Liang, “Blind image deconvolution using a robust GCD approach,” IEEE Trans. Image Process. 8, 295–301 (1999).
[CrossRef]

Lim, J. S.

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, 1990).

Manjunath, B. S.

H. Li, B. S. Manjunath, S. K. Mitra, “Multi-sensor image fusion using the wavelet transform,” in Proceedings of IEEE International Conference on Image Processing (ICIP’94), (Institute of Electrical and Electronics Engineers, 1994), pp. 51–53.

Mateos, J.

M. Vega, J. Mateos, R. Molina, A. K. Katsagegelos, “Bayesian parameter estimation in image reconstruction from sub-sampled blurred observations,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, 2003), pp. 1655–1657.

Milanfar, P.

N. Nguyen, P. Milanfar, G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Process. 10, 573–583 (2001).
[CrossRef]

Millane, R.

Mitra, S. K.

H. Li, B. S. Manjunath, S. K. Mitra, “Multi-sensor image fusion using the wavelet transform,” in Proceedings of IEEE International Conference on Image Processing (ICIP’94), (Institute of Electrical and Electronics Engineers, 1994), pp. 51–53.

Molina, R.

C. A. Segall, R. Molina, A. K. Katsaggelos, “High-resolution images from low-resolution compressed video,” IEEE Signal Process. Mag. 20(3), 37–48 (2003).
[CrossRef]

M. Vega, J. Mateos, R. Molina, A. K. Katsagegelos, “Bayesian parameter estimation in image reconstruction from sub-sampled blurred observations,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, 2003), pp. 1655–1657.

Myers, K. J.

H. H. Barrett, K. J. Myers, Foundations of Image Science (Wiley, 2004), Chap. 15.

Nakagaki, R.

R. Nakagaki, A. K. Katsaggelos, “A VQ-based blind image restoration algorithm,” IEEE Trans. Image Process. 12, 1044–1053 (2003).
[CrossRef]

Ng, M. K.

M. K. Ng, N. K. Bose, “Mathematical analysis of super-resolution methodology,” IEEE Signal Process. Mag. 20, 62–74 (2003).
[CrossRef]

Nguyen, N.

N. Nguyen, P. Milanfar, G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Process. 10, 573–583 (2001).
[CrossRef]

Pai, H. T.

H. T. Pai, A. C. Bovik, “Exact multichannel blind image restoration,” IEEE Signal Process. Lett. 4, 217–220 (1997).
[CrossRef]

Paik, J. K.

J. H. Shin, J. H. Jung, J. K. Paik, “Regularized iterative image interpolation and its application to spatially scalable coding,” IEEE Trans. Consumer Electron. 44, 1042–1047 (1998).
[CrossRef]

Park, M. K.

S. C. Park, M. K. Park, M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21–36 (2003).
[CrossRef]

Park, S. C.

S. C. Park, M. K. Park, M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21–36 (2003).
[CrossRef]

Piella, G.

G. Piella, H. Heijmans, “Multiresolution image fusion guided by a multimodal segmentation,” in Proceedings of ACIVS 2002 (Advanced Concepts for Intelligent Vision Systems), Ghent, Belgium, September2002, pp. 175–182.

Pillai, S. U.

S. U. Pillai, B. Liang, “Blind image deconvolution using a robust GCD approach,” IEEE Trans. Image Process. 8, 295–301 (1999).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, 1991).

Rajan, D.

D. Rajan, S. Chandhuri, M. V. Joshi, “Multi-objective super resolution: concepts and examples,” IEEE Signal Process. Mag. 20(3), 49–61 (2003).
[CrossRef]

Rushforth, C. K.

Salam, B. M.

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Efficient implementation of image interpolation as an inverse problem,” Digital Signal Process. 15, 137–152 (2005).
[CrossRef]

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Sectioned implementation of regularized image interpolation,” in 46th Proceedings of the IEEE Midwest Symposium on Circuits and Systems (Institute of Electrical and Electronics Engineers, 2003), pp. 656–659.
[CrossRef]

S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky, B. M. Salam, F. E. Abd El-Samie, “Optimization of image interpolation as an inverse problem using the LMMSE algorithm,” in Proceedings of 12th IEEE Mediterranean Electrotechnical Conference (Institute of Electrical and Electronic Engineers, 2004), pp. 247–250.
[CrossRef]

Schulz, T. J.

Segall, C. A.

C. A. Segall, R. Molina, A. K. Katsaggelos, “High-resolution images from low-resolution compressed video,” IEEE Signal Process. Mag. 20(3), 37–48 (2003).
[CrossRef]

Sezan, M. I.

P. E. Eren, M. I. Sezan, M. Tekalp, “Robust, object-based high-resolution image reconstruction from low-resolution video,” IEEE Trans. Image Process. 6, 1446–1451 (1997).
[CrossRef] [PubMed]

Shin, J. H.

J. H. Shin, J. H. Jung, J. K. Paik, “Regularized iterative image interpolation and its application to spatially scalable coding,” IEEE Trans. Consumer Electron. 44, 1042–1047 (1998).
[CrossRef]

Sroubek, F.

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the wavelet image fusion process; DWT, discrete wavelet transform.

Fig. 2
Fig. 2

Original undegraded building image and its maximum entropy interpolation: (a) original image and (b) maximum entropy interpolation of the original image.

Fig. 3
Fig. 3

Available observations 5×5 blur operator SNR of 60 dB: (a) observation (1), (b) observation (2), (c) observation (3).

Fig. 4
Fig. 4

Results of the proposed algorithm: (a) fused image and (b) obtained high-resolution image peak SNR of 25.26 dB, CPU of 55 s on a 1 GHz processor.

Fig. 5
Fig. 5

Original undegraded MRI image and its maximum entropy interpolation: (a) original image and (b) maximum entropy interpolation of the original image.

Fig. 6
Fig. 6

Available observations 5×5 blur operator SNR of 60 dB: (a) observation (1), (b) observation (2), (c) observation (3).

Fig. 7
Fig. 7

Results of the proposed algorithm: (a) fused image, and (b) obtained high-resolution image peak SNR of 30.2 dB, CPU of 55 s on a 1 GHz processor.

Equations (41)

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y ( m , n ) = x ( m , n ) * b ( m , n ) + υ ( m , n ) ,
y k ( m , n ) = x ( m , n ) * b k ( m , n ) + υ k ( m , n ) , k = 1 , 2 ,
Y k ( z 1 , z 2 ) = X ( z 1 , z 2 ) B k ( z 1 , z 2 ) + V k ( z 1 , z 2 ) , k = 1 , 2 .
Y k ( z 1 , z 2 ) = X ( z 1 , z 2 ) B k ( z 1 , z 2 ) , k = 1 , 2 .
GCD { B 1 ( z 1 , z 2 ) , B 2 ( z 1 , z 2 ) } = 1 ,
GCD { Y 1 ( z 1 , z 2 ) , Y 2 ( z 1 , z 2 ) } = X ( z 1 , z 2 ) .
Y k { exp [ j ( 2 π m / P ) ] , z 2 } = X { exp [ j ( 2 π m / P ) ] , z 2 } . B k { exp [ j ( 2 π m / P ) ] , z 2 } , k = 1 , 2 .
A ( m , n ) a ( m ) = X { exp [ j ( 2 π m / P ) ] × exp [ j ( 2 π m / P ) ] } .
L ( m , n ) l ( n ) = X { exp [ j ( 2 π m / P ) ] × exp [ j ( 2 π n / P ) ]
A ( m , n ) a ( m ) L ( m , n ) l ( n ) = 0 .
X ( e j ( 2 π m / P ) , e j ( 2 π n / P ) ) = 1 2 [ A ( m , n ) a ( m ) + L ( m , n ) l ( n ) ] .
y k ( m , n ) = x ( m , n ) * b k ( m , n ) + υ k ( m , n ) , k = 1 , 2 , , K .
y K + 1 ( m , n ) = k = 1 K w k y k ( m , n ) ,
k = 1 K w k = 1 .
y K + 1 ( m , n ) = k = 1 K w k [ x ( m , n ) * b k ( m , n ) + υ k ( m , n ) ] .
y K + 1 ( m , n ) = x ( m , n ) * [ k = 1 K w k b k ( m , n ) ] + k = 1 K w k υ k ( m , n ) .
y K + 1 ( m , n ) = x ( m , n ) * b K + 1 ( m , n ) + υ K + 1 ( m , n ) ,
b K + 1 ( m , n ) = [ k = 1 K w k b k ( m , n ) ] ,
υ K + 1 ( m , n ) = [ k = 1 K w k υ k ( m , n ) ] .
B K + 1 ( z 1 , z 2 ) = k = 1 K w k B k ( z 1 , z 2 ) .
B K + 1 ( z 1 , z 2 ) B k ( z 1 , z 2 ) = w 1 B 1 ( z 1 , z 2 ) B k ( z 1 , z 2 ) + w 2 B 2 ( z 1 , z 2 ) B k ( z 1 , z 2 ) + + w k + + w K B K ( z 1 , z 2 ) B k ( z 1 , z 2 ) ,
B i ( z 1 , z 2 ) B k ( z 1 , z 2 ) = Q i ( z 1 , z 2 ) + R i ( z 1 , z 2 ) B k ( z 1 , z 2 ) ,
B K + 1 ( z 1 , z 2 ) B k ( z 1 , z 2 ) = w k + i = 1 i k K w i [ Q i ( z 1 , z 2 ) + R i ( z 1 , z 2 ) B k ( z 1 , z 2 ) ] .
B K + 1 ( z 1 , z 2 ) B k ( z 1 , z 2 ) = Q t ( z 1 , z 2 ) + R t ( z 1 , z 2 ) B k ( z 1 , z 2 ) ,
Q t ( z 1 , z 2 ) = w k + i = 1 i k K w i [ Q i ( z 1 , z 2 ) ] ,
R t ( z 1 , z 2 ) = i = 1 i k K w i R i ( z 1 , z 2 ) 0 .
υ K + 1 ( m , n ) = k = 1 K w k υ k ( m , n )
σ K + 1 2 = k = 1 K w k 2 σ k 2 .
σ K + 1 2 = k = 1 K σ k 2 K 2 .
σ K + 1 2 = σ k 2 K .
x = Df + n ,
D = D 1 D 1 ,
D 1 = 1 2 [ 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 ] .
E = i = 1 p 2 f i log 2 ( f i ) ,
E = f t log 2 ( f ) .
W ( f ) = f t log 2 ( f ) λ [ x Df 2 n 2 ] ,
W ( f ) f = 0 = 1 In ( 2 ) { 1 + In ( f ̂ ) } + λ [ 2 D t ( x D f ̂ ) ] .
In ( f ̂ ) = 1 λ In ( 2 ) 2 D t ( x D f ̂ ) .
f ̂ = exp 1 λ In ( 2 ) 2 D t ( x D f ̂ ) .
f ̂ λ In ( 2 ) 2 D t ( x D f ̂ ) .
f ̂ ( D t D + η I ) 1 D t g ,

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